Multiagent Pathfinding
AAAI Technical Report WS-12-10
Reciprocal Collision Avoidance and
Multi-Agent Navigation for Video Games∗
Jamie Snape, Stephen J. Guy, Ming C. Lin, Dinesh Manocha
Department of Computer Science,
University of North Carolina at Chapel Hill
Chapel Hill, North Carolina 27599
{snape, sjguy, lin, dm}@cs.unc.edu
Jur van den Berg
School of Computing,
University of Utah
Salt Lake City, Utah 84112
berg@cs.utah.edu
Abstract
erties of the system such as separation, alignment, and cohesion of particles. Examples of video games using this
method include Capcom’s Dead Rising (2006) and Ubisoft’s
Assassin’s Creed (2007).
More recently, velocity-based methods (Fiorini and
Shiller 1998) have exhibited improvements in terms of local collision avoidance and behavior of virtual agents, and
improved computational performance, over force-based collision avoidance methods. Rather than using virtual forces to
prevent nearby virtual agents from collisions, velocity-based
methods use the current velocity of each virtual agent in the
group and then extrapolate the position of each virtual agent
for some short time interval under the assumption that the
virtual agent will maintain almost a constant velocity over
some short time interval. Based on predicting the future positions of other virtual agents, each virtual agent tends to
choose an avoiding new velocity based on some optimization. THQ’s video game Warhammer 40,000: Space Marine
(2011) uses a velocity-based approach.
Reciprocal collision avoidance (van den Berg, Lin, and
Manocha 2008) is an extension of the velocity-based approaches. The main difference with prior velocity-based
methods lies in the fact that reciprocal collision avoidance
considers the reciprocity between pairs of virtual agents.
Each virtual agent is assumed to be attempting to avoid a
collision with the other, rather than seeing the other virtual
agent as a moving obstacle. Incorporating reciprocity into
velocity-based approaches typically ensures smoother motion for the virtual agents and may also cause emergent phenomena in groups of virtual agents, such as arching, jamming, bottlenecks, and wake formation (Guy et al. 2010).
Collision avoidance and multi-agent navigation is an important component of modern video games. Recent developments in commodity hardware, in particular the utilization of
multi-core and many-core architectures in personal computers and consoles are allowing large numbers of virtual agents
to be incorporated into game levels in increasing numbers.
We present the hybrid reciprocal velocity obstacle and optimal reciprocal collision avoidance methods for reciprocal
collision avoidance and navigation in video games and described their implementations in C++ as HRVO Library and
RVO2 Library. The libraries can efficiently simulate groups
of twenty-five to one thousand virtual agents in dense conditions and around moving and static obstacles.
1
Introduction
Collision avoidance and navigation among virtual agents is
an important component of modern video games. Recent developments in commodity hardware, in particular the utilization of multi-core and many-core architectures in personal
computers and consoles are allowing large numbers of virtual agents to be incorporated into game levels in increasing
numbers and with increasing fidelity. As a result, there is a
need for efficient techniques to automatically generate realistic behaviors for such groups of virtual agents.
Simple local collision avoidance behaviors, such as flocking (Reynolds 1987), have been implemented using forcebased models in many recent video games and commercial game engines. These methods model groups of virtual
agents as particle systems, with each particle applying a
force on nearby particles. The laws of physics are used to
compute the motion of the particles, along with a set of behaviors specified by game developers that influence prop-
2
2.1
∗
An extended version of this work appeared on the Intel Software Network in March 2012
c 2012, Association for the Advancement of Artificial
Copyright Intelligence (www.aaai.org). All rights reserved.
Local Collision Avoidance
Hybrid Reciprocal Velocity Obstacles
The velocity obstacle (Fiorini and Shiller 1998) of a virtual
agent A, with position pA , induced by a moving obstacle B,
with position pB , in a game level is the set of all velocities
49
(a)
(b)
(c)
(d)
Figure 1: (a) Two virtual agents A and B. (b) The velocity obstacle VOA|B for virtual agent A induced by virtual agent B.
(c) The reciprocal velocity obstacle RVOA |B for virtual agent A induced by virtual agent B. (d) The hybrid velocity obstacle
HRVOA|B for virtual agent A induced by virtual agent B. The current velocity v A is right of the centerline CL, so the left side
of HRVOA|B is the left side of VOA|B and the right side of HRVOA|B is the right side of RVOA|B .
for the virtual agent that will result in a collision between the
virtual agent and the moving obstacle within some short time
interval into the future, assuming that the dynamic obstacle
maintains a constant velocity v B :
virtual agent towards its goal position. If a virtual agent has
to move away from the goal to avoid collisions, then the preferred velocity will differ greatly from the current velocity.
The presence of a third virtual agent may also cause at least
one of the virtual agents to choose a velocity even farther
from its current velocity. This means virtual agents may not
necessarily choose the same side to pass, which may result
in undesirable oscillations known as “reciprocal dances” that
may not resolve quickly and may leave virtual agents effectively deadlocked.
The hybrid reciprocal velocity obstacle (Snape et al.
2011) resolves the problem of reciprocal dances by combining the velocity obstacle and the reciprocal velocity obstacle,
taking one side from each to form a hybrid reciprocal velocity obstacle that is enlarged on one side to discourage virtual
agents from passing each other on different sides. If the velocity of a virtual agent is to the right of the centerline of its
reciprocal velocity obstacle induced by some other virtual
agent, then the virtual agent should choose a velocity to the
right of the reciprocal velocity obstacle. To encourage such
behavior, the reciprocal velocity obstacle is enlarged by replacing the edge on the side that the virtual agents should not
pass, for example, the left side in this case, by the edge of the
corresponding velocity obstacle. If the velocity of the virtual
agent is to the left of the centerline, the procedure is mirrored, exchanging left and right sides. The geometric interpretation of a hybrid reciprocal velocity obstacle HRVOA|B
for a virtual agent A with respect to a virtual agent B, including the location of the centerline and an indication of
the enlarged area, is shown in Figure 1(d).
VOA|B = {v | ∃t > 0 :: t(v−v B ) ∈ D(pB −pA , rA +rB )},
where D(p, r) is an open disc of radius r centered at p,
as shown in Figure 1(b). It follows that if the virtual agent
chooses a velocity within the region corresponding to the velocity obstacle, then the virtual agent and the moving obstacle will potentially collide. If the velocity chosen is outside
the velocity obstacle, then a collision will not occur.
Unfortunately, the velocity obstacle approach does not
work very well for local collision avoidance within a group
of virtual agents where each virtual agent is actively changing its velocity to avoid the other virtual agents, since it
assumes that other virtual agents may not change their velocities. If all virtual agents were to use velocity obstacles
to choose a new velocity, there would be oscillations in the
motion of the virtual agents between successive time steps.
The reciprocal velocity obstacle (van den Berg, Lin, and
Manocha 2008) addresses the problem of oscillations caused
by the velocity obstacle by allowing for the reactive nature of
the other virtual agents. Instead of one virtual agent having
to take all the responsibility for avoiding collisions, reciprocal velocity obstacles let a virtual agent take just half of the
responsibility for avoiding a collision and assume that the
other virtual agent reciprocates by taking care of the other
half:
RVOA|B = {v | 2v − v A ∈ VOA|B },
2.2
where v A is the velocity of virtual agent A, as shown in
Figure 1(c).
The reciprocal velocity obstacle approach guarantees that
if both virtual agents select a velocity outside the reciprocal velocity obstacle induced by the other, and both virtual
agents choose to pass each other on the same side, then the
motion of both robots will be free of collisions and oscillations. However, in order to make progress, the virtual agents
are usually required to select the velocity closest to their preferred velocity, defined as the velocity directed from each
Optimal Reciprocal Collision Avoidance
ORCAτA|B = {v | (v − (v A + 12 u)) · n ≥ 0}.
Optimal reciprocal collision avoidance (van den Berg et
al. 2011) solves the problem of reciprocal dances addressed
by the hybrid reciprocal velocity obstacle in a different way.
This approach augments the velocity obstacle with a halfplane that defines a set of velocities that are both collision
free and will additionally ensure that the motion of the virtual agents will be smooth in all but dense scenarios.
50
(a)
(b)
(c)
Figure 2: (a) Two virtual agents A and B. (b) The truncated velocity obstacle VOA|B for virtual agent A induced by virtual
agent B. (c) The optimal reciprocal collision avoidance half-planes of permitted velocities ORCAA|B for virtual agents A and
B.
The optimal reciprocal collision avoidance half-plane
ORCAA|B for a virtual agent A with respect to a virtual
agent B is defined as follows. As shown in Figure 2(c), let u
be the vector from the relative velocity v A − v B of the virtual agents A and B to the closest point on the boundary of
the truncated velocity obstacle for virtual agent A induced
by virtual agent B:
constraints are relaxed and a new velocity is found using
three-dimensional linear programming (see Section 3.4).
The algorithm used in HRVO Library is broadly similar
except that it uses the ClearPath geometric algorithm (Guy
et al. 2009) to compute new velocities.
3.2
u = ( arg min kv − (v A − v B )k) − (v A − v B ).
τ
v∈∂VOA|B
Both HRVO Library and RVO2 Library choose a new velocity by computing the velocity that is closest to the preferred velocity and is collision free. If the goal position of
the virtual agent is visible, then the preferred velocity is in
the direction of the goal. If the goal position is not visible,
the preferred velocity should be directed to the nearest node
on waypoint graph to the goal or to some point on the nearest edge on a navigation mesh path or a roadmap that leads
to the goal.
Let n be the outward normal of the boundary of the velocity
obstacle at v A − v B + u. It follows that u is the smallest change required to the relative velocity of virtual agents
A and B to avoid a collision. Incorporating reciprocity, each
virtual agent should adjust its velocity by at least 21 u to avoid
the collision. Therefore the velocities permitted by optimal
reciprocal collision avoidance are in a half-plane in the direction of n starting at the point v A + 12 u:
3
3.1
Preferred Velocity
Implementation
3.3
Libraries
Linear Programming
RVO2 Library uses an efficient randomized linear programming algorithm (de Berg et al. 2008) that adds the constraints one by one in random order while keeping track of
the current optimal new velocity for a virtual agent in the
group. Linear programming is an optimization technique,
commonly used in operations research, for finding one specific solution to a set of linear equality and inequality constraints that optimizes a given linear function of the variables. Geometrically, a linear programming algorithm computes a point in a polygon where the function has its maximum or minimum value if such a point exists. A randomized
linear programming algorithm adds the linear constraints in
a random order in order to compute the optimum solution.
For each virtual agent in the group, the randomized linear
programming algorithm has a linear expected running time
with respect to the number of virtual agents that are input
into the algorithm. The algorithm computes the velocity in
that is closest to the preferred velocity of the virtual agent,
and reports failure if the linear program is infeasible.
The hybrid reciprocal velocity obstacle approach and optimal reciprocal collision avoidance have been implemented
as C++ libraries, HRVO Library and RVO2 Library, respectively.
Essentially, the algorithm in RVO2 Library computes the
optimal reciprocal collision avoidance half-planes for a virtual agent induced by the other virtual agents, and then intersects these half-planes to form a region of permitted velocities for the virtual agent. The algorithm then computes the
preferred velocity v pref (see Section 3.2) of the virtual agent
and computes a new velocity v new using two-dimensional
linear programming (see Section 3.3) that is within the region of permitted velocities and as close as possible to the
preferred velocity:
v new
= arg min kv − v pref
A
A k
v∈ORCAτA|B
If there are many virtual agents nearby and there is no
velocity within the region of permitted velocities, then some
51
Figure 3: A screenshot of the benchmark scenario for an integration of RVO2 Library with Unreal Development Kit,
two hundred virtual agents navigating in real time between
randomly chosen locations at the four corners of the game
level.
3.4
Figure 4: A screenshot of the Dungeon scenario included
with Recast and Detour, fifty virtual agents navigating from
one end of the game level to the other on a navigation mesh
using HRVO Library.
brary and RVO2 Library. The libraries can efficiently simulate groups of twenty-five to one thousand virtual agents
in dense conditions and around moving and static obstacles.
RVO2 Library is on average at least twice as fast as HRVO
Library, but HRVO Library results in fewer collisions between virtual agents of RVO2 Library and therefore results
in better local interactions between the virtual agents.
Dense Scenarios
In dense scenarios, when a group of virtual agents is packed
tightly together in part of the game level, there may not be
a velocity that satisfies all the constraints of the linear program and the algorithm would return that the linear program
is infeasible. When this occurs, RVO2 Library computes
the safest possible velocity for the virtual agent, the velocity that minimally penetrates the constraints induced by the
other virtual agents. This can be interpreted geometrically as
moving the edges of the half-planes perpendicularly outward
with equal speed, until exactly one velocity becomes valid.
This velocity may be computed using a three-dimensional
linear program. The same randomized linear programming
algorithm as before may be used by projecting the problem
down onto plane, such that all geometric operations can be
performed in two-dimensions. The three-dimensional linear
program is always feasible, so it always returns a solution.
The running time of the algorithm is still linear with respect
to the number of virtual agents.
3.5
5
References
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Game Engine Integration
RVO2 Library has been integrated into several game engines to either perform local collision avoidance and navigation for groups of virtual agents or improve upon the default implementations provided by the game engine developers. Examples include a multi-platform package for Unity
Technologies’ Unity 3 written in C# and a DLL for Epic
Games’ Unreal Development Kit written in C++ for Microsoft Windows with UnrealScript bindings. A screenshot
from the Unreal Development Kit integration is shown in
Figure 3. An approach broadly similar to the hybrid reciprocal velocity obstacle, as used in HRVO Library, has been
incorporated into the Detour component of the game navigation toolset Recast and Detour to provide local collision
avoidance within a navigation mesh, as shown in Figure 4.
4
Acknowledgments
This work was supported in part by Army Research Laboratory Contract W911NF-04-1-0088, by National Science
Foundation Awards 1000579 and 1117127, and by Intel Corporation.
Conclusion
We have presented the hybrid reciprocal velocity obstacle
and optimal reciprocal collision avoidance methods for reciprocal collision avoidance and navigation in video games
and described their implementations in C++ as HRVO Li-
52